They ended up replacing my barrel. When I received the rifle back with the new barrel installed, it had a considerable amount of lapping compound left behind, so they obviously had to put in some effort to make the required loading force more uniform. It is definitely better now than before, not perfect, but way better. The accuracy is significantly improved as a result. The breech plug also seals quite well on this barrel. There is a section about 8" down the barrel from the muzzle where there appears to be some pitting of some kind, but it doesn't seem to be affecting accuracy. They also sent me some free Fury's, which makes up for what I burned through before, so that was nice.
To others who are looking into buying a Peregrine, I would advise that you chat with someone at Knight, and make clear your expectations about barrel uniformity up front. While I am grateful that they took care of me in the end, this experience has been quite a headache and cost me a lot of precious time. It's pretty clear that Green Mountain has some barrel quality control issues to address, and I'm sure they will over time.
Overall, I still like the rifle.